use the product rule to find the derivative with respect to the independent variable.\n$f(x)=(3x^{3}-1)(2 +…

use the product rule to find the derivative with respect to the independent variable.\n$f(x)=(3x^{3}-1)(2 + 4x^{2})$\n$f(x)=\\square$

use the product rule to find the derivative with respect to the independent variable.\n$f(x)=(3x^{3}-1)(2 + 4x^{2})$\n$f(x)=\\square$

Answer

Explanation:

Step1: Identify (u) and (v)

Let (u = 3x^{3}-1), (v=2 + 4x^{2}).

Step2: Find (u') and (v')

Differentiate (u) with respect to (x): (u'=\frac{d}{dx}(3x^{3}-1)=9x^{2}). Differentiate (v) with respect to (x): (v'=\frac{d}{dx}(2 + 4x^{2})=8x).

Step3: Apply the product rule (y'=u'v+uv')

[ \begin{align*} f'(x)&=(9x^{2})(2 + 4x^{2})+(3x^{3}-1)(8x)\ &=18x^{2}+36x^{4}+24x^{4}-8x\ &=(36x^{4}+24x^{4})+18x^{2}-8x\ &=60x^{4}+18x^{2}-8x \end{align*} ]

Answer:

(60x^{4}+18x^{2}-8x)